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Search: id:A081314
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| A081314 |
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Order of symmetry groups of n points on 3-dimensional sphere with the volume enclosed by their convex hull maximized. |
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+0
3
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| 24, 12, 48, 20, 8, 12, 16, 4, 120, 4, 24, 12, 24, 4, 6, 2, 8, 2, 4, 6, 4, 2, 2, 20
(list; graph; listen)
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OFFSET
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4,1
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COMMENT
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If more than one configuration with maximal volume exists for a given n, the one with the largest symmetry group is chosen. Berman and Hanes give optimality proofs for n<=8. Higher terms are only conjectures. An independent verification of the results by Hardin, Sloane and Smith has been performed by Pfoertner in 1992 for n<28. An archive of the results with improvements for n=23,24 added in 2003 is available at link. A conjectured continuation of the sequence starting with n=28 is: 12,6,2,6,120,2,4,4,2,20,4,12,24,12,20,4,8,2,2,2,4,1,24
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REFERENCES
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Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970)
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LINKS
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R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes
Hugo Pfoertner, Maximal Volume Arrangements of Points on Sphere. Visualizations for n<=21.
Hugo Pfoertner, Maximal Volume Arrangements: Archive
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EXAMPLE
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a(12)=120 because the order of the point group of the icosahedron, which is also the best known arrangement for the maximal volume problem is 120. a(7)=20 because the double 7-pyramid proved optimal by Berman and Hanes has dihedral symmetry order 20.
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CROSSREFS
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Number of distinct edges in convex hull: A081366. Symmetry groups for Tammes problem: A080865.
Sequence in context: A079341 A080865 A040555 this_sequence A119872 A002550 A075605
Adjacent sequences: A081311 A081312 A081313 this_sequence A081315 A081316 A081317
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KEYWORD
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hard,nonn
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AUTHOR
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Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 19 2003
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